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Theorem supisoti 6865
Description: Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
Hypotheses
Ref Expression
supiso.1  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
supiso.2  |-  ( ph  ->  C  C_  A )
supisoex.3  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )
supisoti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
Assertion
Ref Expression
supisoti  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Distinct variable groups:    v, u, x, y, z, A    u, C, v, x, y, z    ph, u    u, F, v, x, y, z    u, R, x, y, z    u, S, v, x, y, z   
u, B, v, x, y, z    v, R    ph, v, x
Allowed substitution hints:    ph( y, z)

Proof of Theorem supisoti
Dummy variables  w  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supisoti.ti . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
21ralrimivva 2491 . . . . . 6  |-  ( ph  ->  A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )
3 supiso.1 . . . . . . 7  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
4 isoti 6862 . . . . . . 7  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  <->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) ) )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  <->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) ) )
62, 5mpbid 146 . . . . 5  |-  ( ph  ->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
76r19.21bi 2497 . . . 4  |-  ( (
ph  /\  u  e.  B )  ->  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
87r19.21bi 2497 . . 3  |-  ( ( ( ph  /\  u  e.  B )  /\  v  e.  B )  ->  (
u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
98anasss 396 . 2  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u  =  v  <-> 
( -.  u S v  /\  -.  v S u ) ) )
10 isof1o 5676 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  F : A -1-1-onto-> B
)
11 f1of 5335 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
123, 10, 113syl 17 . . 3  |-  ( ph  ->  F : A --> B )
13 supisoex.3 . . . 4  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )
141, 13supclti 6853 . . 3  |-  ( ph  ->  sup ( C ,  A ,  R )  e.  A )
1512, 14ffvelrnd 5524 . 2  |-  ( ph  ->  ( F `  sup ( C ,  A ,  R ) )  e.  B )
161, 13supubti 6854 . . . . . 6  |-  ( ph  ->  ( j  e.  C  ->  -.  sup ( C ,  A ,  R
) R j ) )
1716ralrimiv 2481 . . . . 5  |-  ( ph  ->  A. j  e.  C  -.  sup ( C ,  A ,  R ) R j )
181, 13suplubti 6855 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  A  /\  j R sup ( C ,  A ,  R )
)  ->  E. z  e.  C  j R
z ) )
1918expd 256 . . . . . 6  |-  ( ph  ->  ( j  e.  A  ->  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R
z ) ) )
2019ralrimiv 2481 . . . . 5  |-  ( ph  ->  A. j  e.  A  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R
z ) )
21 supiso.2 . . . . . . 7  |-  ( ph  ->  C  C_  A )
223, 21supisolem 6863 . . . . . 6  |-  ( (
ph  /\  sup ( C ,  A ,  R )  e.  A
)  ->  ( ( A. j  e.  C  -.  sup ( C ,  A ,  R ) R j  /\  A. j  e.  A  (
j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R
z ) )  <->  ( A. w  e.  ( F " C )  -.  ( F `  sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) ) ) )
2314, 22mpdan 417 . . . . 5  |-  ( ph  ->  ( ( A. j  e.  C  -.  sup ( C ,  A ,  R ) R j  /\  A. j  e.  A  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R z ) )  <-> 
( A. w  e.  ( F " C
)  -.  ( F `
 sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) ) ) )
2417, 20, 23mpbi2and 912 . . . 4  |-  ( ph  ->  ( A. w  e.  ( F " C
)  -.  ( F `
 sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) ) )
2524simpld 111 . . 3  |-  ( ph  ->  A. w  e.  ( F " C )  -.  ( F `  sup ( C ,  A ,  R ) ) S w )
2625r19.21bi 2497 . 2  |-  ( (
ph  /\  w  e.  ( F " C ) )  ->  -.  ( F `  sup ( C ,  A ,  R
) ) S w )
2724simprd 113 . . . 4  |-  ( ph  ->  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) )
2827r19.21bi 2497 . . 3  |-  ( (
ph  /\  w  e.  B )  ->  (
w S ( F `
 sup ( C ,  A ,  R
) )  ->  E. k  e.  ( F " C
) w S k ) )
2928impr 376 . 2  |-  ( (
ph  /\  ( w  e.  B  /\  w S ( F `  sup ( C ,  A ,  R ) ) ) )  ->  E. k  e.  ( F " C
) w S k )
309, 15, 26, 29eqsuptid 6852 1  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394    C_ wss 3041   class class class wbr 3899   "cima 4512   -->wf 5089   -1-1-onto->wf1o 5092   ` cfv 5093    Isom wiso 5094   supcsup 6837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-sup 6839
This theorem is referenced by:  infisoti  6887  infrenegsupex  9357  infxrnegsupex  11000
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